It’s one of the most complex yet important theorems in probability. Bayes’ theorem can determine the probability of something is true based on given facts in relation to a given event. While it might be easy for some to understand the concept through textbook explanations, what does Bayes’ theorem look like in a real-world application? Derek Muller from the YouTube educational group Veritasium decided to break it down.
The theorem gets its name from Reverend Thomas Bayes, an 18th-century scholar. He wanted to create an equation that accounted for updated beliefs and information. Sir Harold Jeffreys later said that the Bayes’ theorem “is to the theory of probability what the Pythagorean theorem is to geometry.”
Simply put, A and B are events and P(B) cannot equal 0. But what does that look like in a modern example? Derek uses several ways to think about this theorem given various information. He first starts with using a hypothetical of contracting an illness. You’re not feeling well, so you see a doctor for some tests. The doctor returns to tell you that you’ve tested positive as possibly having a particularly nasty disease that affects only 1 percent of the population. The tests are 99 percent positive at accurately diagnosing those that had the disease. It only incorrectly diagnoses 1 percent of the population that doesn’t have the disease. For most people, they would assume they’d have a 99 percent chance of suffering that disease. However, the odds are more hopeful when using Bayes’ theorem into account.
Bayes’ theorem was intended to be evolved and expanded. Bayes himself developed the formula while adding new information during his own personal tests.
So other than theoretical applications about a diagnosis, how else is the theorem used? Well, a type of the theorem is used to filter out spam from your email. It takes into account words, how often they’re found and their relation to spam accounts while constantly adjusting for each new spam email.
The best part of the video comes at 7:30 when Derek starts discussing the psychological applications of the theorem. He eloquently explains just how theorems like this one and the logic behind them can inadvertently weave their way into the public psyche, even unnamed.